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The ability to switch between various representations is an invaluable problem-solving skill in physics. In addition, research has shown that using multiple representations can greatly enhance a person's understanding of mathematical and physical concepts. This paper describes a study of student difficulties regarding interpreting, constructing, and switching between representations of vector fields, using both qualitative and quantitative methods. We first identified to what extent students are fluent with the use of field vector plots, field line diagrams, and symbolic expressions of vector fields by conducting individual student interviews and analyzing in-class student activities. Based on those findings, we designed the Vector Field Representations test, a free response assessment tool that has been given to 196 second-and third-year physics, mathematics, and engineering students from four different universities. From the obtained results we gained a comprehensive overview of typical errors that students make when switching between vector field representations. In addition, the study allowed us to determine the relative prevalence of the observed difficulties. Although the results varied greatly between institutions, a general trend revealed that many students struggle with vector addition, fail to recognize the field line density as an indication of the magnitude of the field, confuse characteristics of field lines and equipotential lines, and do not choose the appropriate coordinate system when writing out mathematical expressions of vector fields.

IN CONTRAST TO the usual measure of "demand" according to some objective criteria of "need," reserves are assumed in this study to result from the utility-maximizing behavior of the official holders. Governments are hypothesized to hold reserves in order to reduce income fluctuations otherwise induced by exogenous changes in external demand. The less willing is a government to have such changes spill over to the domestic economy, the higher the average reserves it must hold. But these holdings involve an opportunity cost: the income foregone thereby. There is thus a tradeoff between the reduction in income variation and the reduction in income level implicit in any average level of reserves. Optimum reserves can be derived given knowledge of the government's utility function, the distribution of exogenous changes in demand, the link between domestic income and the foreign sector (via monetary policy and commercial policy, the latter defined broadly to include all measures affecting the choices between domestic and foreign commodities and debt instruments), and the opportunity cost of reserves. A model specifying the interrelationships among these variables is developed and an equation for optimum reserve holdings is derived therefrom.The hypothesis is tested against annual data for 46 countries in the postwar period. Linear and logarithmic regressions using the pooled data yield coefficients relating average reserves to the standard deviations of exports, the level of income, and three alternative measures of opportunity cost (per capita income, the ratio of capital flows to income, and the balance of international indebtedness). All are significant and have the expected signs with the exception of some opportunity-cost proxies. Alternative policy options are represented by exchange-rate changes, the convertibility status of the currency, the relation between changes in exports and changes in reserves, and the risk of reserve depletion implicit in actual holdings. These have low or zero significance, suggesting that balance-of-payments policies form a simultaneous "package." Dummy variables are included to test for intercountry differences apart from those explained by the independent variables. Six to eight of the countries have reserves significantly above or below average in the log form. Separate tests on moreopen and less-open countries and more-developed and less-developed countries reveal significant differences in the coefficients of the independent variables. While export fluctuations have roughly the same impact in all subgroups, the less-developed and less-open countries have a higher income elasticity of demand. A further test on annual cross sections confirms the hypothesis that the coefficients did not change significantly in the period 1953 to 1965.The implications of a stable demand function for international monetary reform are discussed in the final chapter. Given a commitment to fixed exchange rates, any supply policy that does not consider the reactions of reserves holders would lead to secula...

We report on the design of an introductory thermal physics module taught through Problem Based Learning ͑PBL͒ within a lecture-based curriculum and discuss how some of the potential benefits of PBL, in particular, effective mixed-ability teaching and increased student motivation, can be realized within such a framework. We describe how the transition from lecture-based to PBL teaching has taken place and illustrate the development and implementation of our methodology with two problems from the module.

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